As we said, things can get very complicated. There aren't any simple formulas
to decide how an election will turn out once some of the voters start voting
If the two voters represented in column 4 (they prefer mushrooms) figure
out that you are going to switch your vote to sausage making it the winner,
they may switch their votes to anchovies. Now anchovies win 6 to 5. Why
should the voters switch? Remember, anchovies are their second choice and
are preferable to sausage.
Now suppose you see that the column 4 voters will switch to anchovies.
You might get together with the voters in column 2 and talk to the voters
in column 4. Tell them that if they don't switch to anchovies but
stick with mushrooms then the voters in columns 1 and 2 will also vote
for mushrooms. Now mushrooms beat anchovies 7 to 4. The voters in column
4 get their first choice and at least the voters in columns 1 and 2 don't
have to eat anchovies.
But suppose the voters in columns 1 and 2 are lying about switching to
mushrooms. Once they convince column 4 to stick with mushrooms they could
go back and vote for sausage. Sausage wins again.
But if column 4 voters guess that columns 1 and 2 are are lying ...
happens if we try the run-off method?